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David Roberts
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This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc. 13 (2000),773--806.

This is really a remarkable and beautiful theorem.

This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc. 13 (2000),773--806. This is really a remarkable and beautiful theorem.

This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See

This is really a remarkable and beautiful theorem.

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Douglas Lind
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This question is completely answered, and the result is that the condition involving the Moebius inversion you mention is both necessary and sufficient! See K. H. Kim, N. Ormes, F. Roush. The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc. 13 (2000),773--806. This is really a remarkable and beautiful theorem.